Cutting mesh by projecting texture
This can be done using the Displace modifier.


Subdivide the circle a few times to ensure it has enough polygons to displace
UV unwrap the circle and apply the texture to it, ensuring it fits nicely to the circle
Add a Displace modifier, using the texture, and selecting the correct UV map in the modifier settings, this should give you something like this (I used your example image, but the original will probably look better)

Apply all modifiers and go into edit mode, then select all vertices that are not on the ground plane and delete them. (press B to select vertices and then press Numpad 1 to go into side view, also you'll want to turn Limit selection to visible off, which can be found on the header of the 3d view at the bottom)

Cleanup the result by selecting all vertices and pressing S Z 0 Enter to make all resulting vertices flat to the ground. Also you could add  a Decimate modifier, or cleanup a bit manually.